Statistical Inference For RCINAR(1) Model Based On Negative Binomial Thinning Operator

In this paper, we introduce a new integer valued time series model, we call the following equation RCINAR(l)process based on negative binomial thinning operator:where{α_t}is i.i.d. random variable,α_t∈B(?) R⁺,with distribution functionP_αt;ε_t are i.i.d. positive integer valued r.v. with probability function f_ε,and X₀,{α_t},{ε_t} are?independent.α_t * X_t-1 = (?), where W_i^(t) is i.i.d.geometric(α_t/(l+α_t)),withprobability mass function given by P(W_i^(t) = (?), x = 0,1,…furthermore, W_i^(s) and W_j^(t)(s≠t)are independent,i, j= 1, 2,….Letα= E(α_t),σ_α² = Var(α_t),μ_ε= E(ε_t),σ_ε² = Var(ε_t), assume they are all finite.The process which is given by(1) is a Markov chain, the transition probability is as follows:Proposition 1. {X_t} is a Markov onΩ={0,1,…},the transition probability isSince the Moments and Conditional Moment are useful in deriving the parameters estimation equation, so we first give the following proposition.Proposition 2.For t≥1, we havewhere (?) Theorem 1. Suppose that0 <σ_α² +α² < 1, then(1)uniquely exit a strictly stationary and ergodic solution.For model (1), our main interest lies in estimating the parameterα= E(α_t) andμ_ε= E(ε_t),in the following section, we suppose {X_t} is a strictly stationary and ergodic solution of (1), we mainly consider two method named CLS and YW mcthod.An advantage of these methods is that they do not require specifying the exact family of distributions for the model.then CLS is defined asTheorem2. SupposeE|X_t|⁴ <∞.then under regular condition(Klimo and Neison), (?) is consistent, andwhere,LetFrom Proposition2(vi),we haveα= (?), where (?) = (?)γ₁ = Cov(X_t,X_t-1),γ₀ = Var(X_t), then the Yule-Walker estimator forαisunder (?) ,we define where (?)Theorem3.Letα= E(α_t),μ_ε= E(ε_t), then (?), (?) is the strongly consistent estimator forα, where (?), (?) is defined in(2),(3).At last, we have some simulation results, from which we can see that CLS and YW estimators are very close, both methods are acceptable. When n = 1500, CLS and YW estimators are the same.